Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

Abstract

Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We review here several properties of the exponential manifold on a suitable set E of mutually absolutely continuous densities. We study in particular the fine properties of the Kullback-Liebler divergence in this context. We also show that this setting is well-suited for the study of the spatially homogeneous Boltzmann equation if E is a set of positive densities with finite relative entropy with respect to the Maxwell density. More precisely, we analyze the Boltzmann operator in the geometric setting from the point of its Maxwell's weak form as a composition of elementary operations in the exponential manifold, namely tensor product, conditioning, marginalization and we prove in a geometric way the basic facts, i.e., the H-theorem. We also illustrate the robustness of our method by discussing, besides the Kullback-Leibler divergence, also the property of Hyvärinen divergence. This requires us to generalize our approach to Orlicz-Sobolev spaces to include derivatives.

A class of non-parametric statistical manifolds modelled on Sobolev space

“…One way of achieving this is to use model spaces of Sobolev type. This is carried out in the context of the exponential Orlicz manifold in [17], where it is applied to the spatially homogeneous Boltzmann equation. Manifolds modelled on the Banach spaces C k b (B; R), where B is an open subset of an underlying (Banach) sample space, are developed in [28], and manifolds modelled on Fréchet spaces of smooth densities are developed in [4,7] and [28].…”

Information Geometry of the Gaussian Space

“…We conclude this section by recalling the following tensor property of the exponential space and of the mixture space, see [10].…”

“…We proceed in this section to the extension of our discussion of translation statistical models to statistical models of the Gaussian space whose densities are differentiable. We restrict to generalities and refer to previous work in [10] for examples of applications, such as the discussion of Hyvärinen divergence. This is a special type of divergence between densities that involves an L 2 -distance between gradients of densities [9] which has multiple applications.…”

Choosing between Higher Moment Maximum Entropy Models and Its Application to Homogeneous Point Processes with Random Effects

“…Such a comparison can be done by evaluating how close each predictive density f p (y|x) is to the true density f (y|x; θ), where θ is a vector of unknown parameters. To judge the goodness-of-fit of a given predictive method [23][24][25], a common approach has been to assess the relative closeness with the average Kullback-Leibler (KL) divergence [26], which is defined by…”